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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Abstract parabolic problems
with critical nonlinearities and applications
to Navier-Stokes and heat equations


Authors: José M. Arrieta and Alexandre N. Carvalho
Journal: Trans. Amer. Math. Soc. 352 (2000), 285-310
MSC (1991): Primary 34G20, 58D25; Secondary 35K05, 35Q30
Published electronically: September 21, 1999
MathSciNet review: 1694278
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Abstract: We prove a local existence and uniqueness theorem for abstract parabolic problems of the type $\dot x=Ax+f(t,x)$ when the nonlinearity $f$ satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.


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Additional Information

José M. Arrieta
Affiliation: Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: arrieta@sunma4.mat.ucm.es

Alexandre N. Carvalho
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, C.P. 668, São Carlos, SP. Brazil
Email: andcarva@icmsc.sc.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02528-3
PII: S 0002-9947(99)02528-3
Keywords: Abstract parabolic equations, critical nonlinearities, growth conditions, local existence, uniqueness, Navier-Stokes, heat equations.
Received by editor(s): August 6, 1997
Published electronically: September 21, 1999
Additional Notes: The first author’s research was partially supported by FAPESP-SP-Brazil, grant # 1996/3289-4. The second author’s research was partially supported by CNPq-Brazil, grant # 300.889/92-5
Article copyright: © Copyright 1999 American Mathematical Society